Math 20-1 Chapter 8 Systems of Equations

8.1 Solving Systems Graphically

Teacher Notes

8.1 Linear-Quadratic System of EquationsA Linear-Quadratic System of Equations is a linear equation and a quadratic equation involving the same two variables. The solution(s) to this system are the point(s) on the graph where the line intersects the parabola (if it does at all).

A Quadratic-Quadratic System of Equations is two quadratic equations involving the same variables. The solution(s) to this system are the point(s) on the graph where the two parabolas intersect (if they do at all).

8.1.1

2

2

4

y x

y x

2

2

2

4

y x

y x

Consider the sketch of a line and a parabola

a) What is the maximum number of intersection points that a line and aparabola could have? Illustrate with a diagram.

b) What is the minimum number of intersection points that a line and aparabola could have? Illustrate with a diagram.

c) Is it possible for a line and a parabola to not to intersect? Illustrate with a diagram.

Linear-Quadratic Systems

There could be two points of intersection.

There could be one point of intersection.

Yes, the line could be outside of the parabola.8.1.2

Determine the solutions for each of the following systems

Linear-Quadratic Systems

no solution

(–1,–4) and (3, 0)

(–4,–6) and (0,–2)

(5, 4)

8.1.3

When Solving GraphicallyExplain Why….

1 7x

2

2

4

y x

y x

The solution isx = -8 or x = 6

The solution is(-2, 0) or (3, 5)

Solve a System of Linear-Quadratic Systems

Solve the following system of equations graphically.

2 4 1

7 9

y x x

y x

From the graph the point of intersection is (-2, -5).

8.1.4

or (5, 44).

Solve a System of Linear-Quadratic Systems

2 4 1

7 9

y x x

y x

Verify the solution by substituting in to the original equations

For (–2, –5)

y = x2 + 4x – 1–5 = (–2)2 + 4(–2) – 1–5 = 4 – 8 – 1–5 = –5

y = 7x + 9–5 = 7(–2) + 9–5 = –14 + 9–5 = –5

8.1.5

Your TurnSolve the system

2 5 2

2

y x x

y x

From the graph the points of intersection are (–4, –6) and (0, –2).

8.1.6

The price C, in dollars per share, of a high-tech stock has fluctuated over a twelve-year period according to the equation C = 14 + 12x - x2 , where x is in years. The price C, in dollars per share, of a second high-tech stock has shown a steady increase during the same time period according to the relationship C = 2x + 30. For what values are the two stock prices the same?

Solve a System of Linear-Quadratic Systems

Graph the system of equations

C = 14 + 12x - x2 C = 2x + 30

From the graph, the two stock prices are the same at 2 years at $34 per share and at 8 years at $46 per share.

8.1.7

QUADRATIC-QUADRATIC SYSTEMS

Is it possible for a quadratic-quadratic system to have an infinite number of solutions?

Quadratic-Quadratic Systems

no solution

one solution

Determine the number of solutions for each of the following systems.

two solutions

two solutions

8.1.9

Quadratic-Quadratic Systems

Solve the following system of equations graphically.

2

2

2 8 7 0

4 2 0

x x y

y x x

From the graph the points of intersection are (1, 1) and (3, 1).8.1.10

Solve a System of Quadratic-Quadratic Equations

Verify the solution by substituting in to the original equations

2

2

2 8 7 0

4 2 0

x x y

y x x

For (1, 1)

2

2

2

2

2 8 7 0

2( ) 8( ) 7 0

2 8 7 1 0

0 0

4 2 0

( ) 4

1 1 1

1 ( ) 2 01 1

1 1 4 2 0

0 0

x x y

y x x

For (3, 1)

2

2

2

2

2 8 7 0

2( ) 8( ) 7 0

18 24 7 1 0

0 0

4 2 0

( ) 4( ) 2 0

1 9 12 2 0

0 0

3 3 1

1 3 3

x x y

y x x

8.1.11

Solve a System of Quadratic-Quadratic Equations Graphically

Your TurnSolve the system

2

2

6 1

4 4 6

x x y

x x y

From the graph the points of intersection are (–2.5, 41) and (1, 6).8.1.12

Andrea and Erin have joined in on a game of pickup baseball. Andrea hits the baseball that travels on a path modeled by the equation h = –0.006x2 + 0.54x + 1.4, where h is the height of the ball above the ground in metres and x is the horizontal distance from home plate in metres.Erin is in the outfield directly in line with the path of the ball. She runs and jumps, trying to catch the ball. Her jump is modeled by the equation h = –0.19x2 + 35.9x – 1690. Determine the height when the ball is caught and its distance from home plate.

Solve a System of Quadratic-Quadratic Equations Graphically

The ball was caught at a height of 1.54 m and a distance of 89.7 m from home plate.

8.1.13

Determine the value(s) for b that would result in the linear-quadratic system y = x2 + 3x + 1 and y = –x – b having a) One point of intersection.b) Two points of intersection.c) No points of intersection

Solve a System of Linear-Quadratic Systems

a) For one point of intersection, b = 3.

b) For two points of intersection, b < 3.

c) For no points of intersection, b > 3.

Use the gsp file to manipulate the line y = –x – b to determine the points of intersection.

8.1.8

How can we determine these values algebraically?

Determine the value(s) for b that would result in the linear-quadratic system y = x2 + 3x + 1 and y = –x – b having a) One point of intersection.b) Two points of intersection.c) No points of intersection

Algebraically

The solution is where the two expressions are equal.

2 3 1x x x b 2 4 1 0x x b

Use the discriminant

b2 - 4ac > 0

b2 - 4ac < 0

b2 - 4ac = 0

24 4 1 1 0b

Assignment

Suggested QuestionsPage 435:1, 3, 4a,c,d, 5a,e, 7b, 8, 10, 13, 14